Extending Beran (1982) bootstrap distribution-function refinements to high dimensions

Extend Beran’s (1982) fixed-dimensional results on bootstrap estimation error for distribution functions (in Kolmogorov distance) to the high-dimensional setting for maxima of sums of independent random vectors, providing a formal explanation for the observed superiority of bootstrap over normal approximation.

Background

The paper notes empirical evidence that bootstrap can outperform normal approximation in estimating the distribution function of high-dimensional maxima, paralleling known fixed-dimensional results by Beran (1982).

A high-dimensional generalization would formalize this phenomenon and complement the paper’s results on coverage probabilities by addressing distribution-function approximation directly.